Chaotic dynamics in periodic potentials

Detalhes bibliográficos
Autor(a) principal: Lazarotto, Matheus Jean
Data de Publicação: 2023
Tipo de documento: Tese
Idioma: eng
Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
Texto Completo: https://www.teses.usp.br/teses/disponiveis/43/43134/tde-02052023-144634/
Resumo: Spatial diffusion of particles in periodic potential models has provided a good framework for studying the role of chaos in global properties of classical systems. Here a square-symmetric potential, classically modeled from an optical lattice Hamiltonian system, was initially used to numerically study diffusion transitions under variation of the control parameters. Sudden transitions between normal and ballistic regimes were found and characterized by inspection of topological changes taking place in phase-space. Particular transitions, correlated with increases in global stability area, were seen to occur for energy levels where local maxima points become accessible, deviating trajectories approaching them. These instabilities promote a slowing down of the dynamics and an island myriad bifurcation phenomenon, along with the suppression of long flights within the lattice. On further investigating the island myriad, its structure was found to be intimately related to the translational and rotational symmetries of the lattice potential. With a high fractal pattern, the myriad of islands is concentrically organized in isochronous chains, formed either by orbits with limited range or high escape transport. As the local maxima points change with the control parameters, the bifurcation of each chain sequentially follows a separatrix reconnection, as in a local non-twist scenario. Due to the myriads dependence on the tiling symmetry property of the square lattice, its presence was conjectured and confirmed also for a hexagonal lattice, although found in attenuated form due to extra instabilities in the potential. Beyond that, the numerical techniques applied for analyses along this work are of wide use and can be adapted to generic conservative systems, allowing their study as their parameters change in an automated way.
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spelling Chaotic dynamics in periodic potentialsDinâmica caótica em potenciais periódicosCaos HamiltonianoDiffusionDifusãoDynamical SystemsHamiltonian chaosPeriodic potentialPotencial periódicoSistemas dinâmicosSpatial diffusion of particles in periodic potential models has provided a good framework for studying the role of chaos in global properties of classical systems. Here a square-symmetric potential, classically modeled from an optical lattice Hamiltonian system, was initially used to numerically study diffusion transitions under variation of the control parameters. Sudden transitions between normal and ballistic regimes were found and characterized by inspection of topological changes taking place in phase-space. Particular transitions, correlated with increases in global stability area, were seen to occur for energy levels where local maxima points become accessible, deviating trajectories approaching them. These instabilities promote a slowing down of the dynamics and an island myriad bifurcation phenomenon, along with the suppression of long flights within the lattice. On further investigating the island myriad, its structure was found to be intimately related to the translational and rotational symmetries of the lattice potential. With a high fractal pattern, the myriad of islands is concentrically organized in isochronous chains, formed either by orbits with limited range or high escape transport. As the local maxima points change with the control parameters, the bifurcation of each chain sequentially follows a separatrix reconnection, as in a local non-twist scenario. Due to the myriads dependence on the tiling symmetry property of the square lattice, its presence was conjectured and confirmed also for a hexagonal lattice, although found in attenuated form due to extra instabilities in the potential. Beyond that, the numerical techniques applied for analyses along this work are of wide use and can be adapted to generic conservative systems, allowing their study as their parameters change in an automated way.A difusão espacial de partículas em potenciais periódicos têm fornecido bons cenários para o estudo do papel do caos em propriedades globais na dinâmica de sistemas clássicos. Neste trabalho, um potencial de simetria quadrada, classicamente modelado a partir de um Hamiltoniano de rede óptica, foi inicialmente usado para o estudo de transições de difusão conforme a variação dos parâmetros de controle. Transições repentinas entre os regimes normal e balístico de difusão foram encontradas e descritas em termos das mudanças topológicas acontecendo no espaço de fase. Em particular, transições correlacionadas com aumentos na área de estabilidade foram vistas para níveis de energia onde máximos locais do potencial tornam-se acessíveis. Estas instabilidades promovem uma desaceleração da dinâmica e um fenômeno de miríade de ilhas, assim como a supressão de voos longos na rede. Ao se investigar o fenômeno de miríade em detalhe, sua estrutura foi vista ser intimamente ligada às simetrias translacional e rotacional do potencial da rede. Com alta fractalidade, a miríade de ilhas organiza-se em camadas concêntricas de cadeias isócronas, formadas tanto por órbitas de alcance limitado ou com alto transporte para escape. Conforme os pontos de máximo local variam com os parâmetros de controle, as cadeias de ilha sequencialmente bifurcam seguindo reconexões de separatriz, em um cenário local não-twist. Devido à dependência com a simetria de preenchimento da rede quadrada, a presença da miríade foi conjecturada e confirmada também para uma rede hexagonal, porém, em forma atenuada devido à fontes extras de instabilidade do potencial. Além destes resultados, os métodos numéricos aplicados em análises ao longo deste trabalho são de ampla aplicação e adaptáveis à sistemas dinâmicos genéricos, permitindo a automação de seu estudo conforme seus parâmetros de controle mudam.Biblioteca Digitais de Teses e Dissertações da USPCaldas, Ibere LuizElskens, YvesLazarotto, Matheus Jean2023-04-18info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/43/43134/tde-02052023-144634/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2023-08-21T15:32:45Zoai:teses.usp.br:tde-02052023-144634Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212023-08-21T15:32:45Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false
dc.title.none.fl_str_mv Chaotic dynamics in periodic potentials
Dinâmica caótica em potenciais periódicos
title Chaotic dynamics in periodic potentials
spellingShingle Chaotic dynamics in periodic potentials
Lazarotto, Matheus Jean
Caos Hamiltoniano
Diffusion
Difusão
Dynamical Systems
Hamiltonian chaos
Periodic potential
Potencial periódico
Sistemas dinâmicos
title_short Chaotic dynamics in periodic potentials
title_full Chaotic dynamics in periodic potentials
title_fullStr Chaotic dynamics in periodic potentials
title_full_unstemmed Chaotic dynamics in periodic potentials
title_sort Chaotic dynamics in periodic potentials
author Lazarotto, Matheus Jean
author_facet Lazarotto, Matheus Jean
author_role author
dc.contributor.none.fl_str_mv Caldas, Ibere Luiz
Elskens, Yves
dc.contributor.author.fl_str_mv Lazarotto, Matheus Jean
dc.subject.por.fl_str_mv Caos Hamiltoniano
Diffusion
Difusão
Dynamical Systems
Hamiltonian chaos
Periodic potential
Potencial periódico
Sistemas dinâmicos
topic Caos Hamiltoniano
Diffusion
Difusão
Dynamical Systems
Hamiltonian chaos
Periodic potential
Potencial periódico
Sistemas dinâmicos
description Spatial diffusion of particles in periodic potential models has provided a good framework for studying the role of chaos in global properties of classical systems. Here a square-symmetric potential, classically modeled from an optical lattice Hamiltonian system, was initially used to numerically study diffusion transitions under variation of the control parameters. Sudden transitions between normal and ballistic regimes were found and characterized by inspection of topological changes taking place in phase-space. Particular transitions, correlated with increases in global stability area, were seen to occur for energy levels where local maxima points become accessible, deviating trajectories approaching them. These instabilities promote a slowing down of the dynamics and an island myriad bifurcation phenomenon, along with the suppression of long flights within the lattice. On further investigating the island myriad, its structure was found to be intimately related to the translational and rotational symmetries of the lattice potential. With a high fractal pattern, the myriad of islands is concentrically organized in isochronous chains, formed either by orbits with limited range or high escape transport. As the local maxima points change with the control parameters, the bifurcation of each chain sequentially follows a separatrix reconnection, as in a local non-twist scenario. Due to the myriads dependence on the tiling symmetry property of the square lattice, its presence was conjectured and confirmed also for a hexagonal lattice, although found in attenuated form due to extra instabilities in the potential. Beyond that, the numerical techniques applied for analyses along this work are of wide use and can be adapted to generic conservative systems, allowing their study as their parameters change in an automated way.
publishDate 2023
dc.date.none.fl_str_mv 2023-04-18
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
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dc.identifier.uri.fl_str_mv https://www.teses.usp.br/teses/disponiveis/43/43134/tde-02052023-144634/
url https://www.teses.usp.br/teses/disponiveis/43/43134/tde-02052023-144634/
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv
dc.rights.driver.fl_str_mv Liberar o conteúdo para acesso público.
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Liberar o conteúdo para acesso público.
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
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dc.publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
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reponame:Biblioteca Digital de Teses e Dissertações da USP
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